Indefinite Integrals#
The Antiderivative#
Definition
A function \(F\) is an antiderivative of \(f\) on an interval \(I\) if
\[F'(x) = f(x)\]
for all \(x\) in \(I\).
In other words, the phrase “\(F\) is an antiderivative of \(f\)” means the same thing as the phrase “\(f\) is the derivative of \(F\).”
Example 1#
Show that \(F(x) = 3x^5 + 4x^3 + 7x\) is an antiderivative of \(f(x) = 15x^4+12x^2+7\).
Step 1: Verify that \(F'(x) = f(x)\).
\[\begin{align*}
F'(x) &= \frac{d}{dx}\left( 3x^5 + 4x^3 + 7x \right) \\
&= 3 \cdot 5 x^4+ 4\cdot 3x^2 + 7\\
& = 15x^4+12x^2+7 \\
& = f(x)
\end{align*}\]
Because \(F'(x) = f(x)\), \(F\) is an antiderivative of \(f\).
The Indefinite Integral#
Definition
The indefinite integral of \(f(x)\) with respect to \(x\), denoted
\[\int f(x) ~dx\]
represents the most general antiderivative of \(f\).
In other words, if \(F\) is an antiderivative of \(f\), then
\[\int f(x) ~dx = F(x)+C.\]